The Place of Exceptional Covers Among all Diophantine Relations

Let Fq be the finite field of order q. We use this characterization of exceptional covers (finite flat morphism; of quasi-projective normal varieties) over Fq: A cover maps one-one on Fqt points for -ly many t. Lenstra suggested there may be an Exceptional (cover) Tower. We construct it and its canonical limit group, permutation representation pair. The essence of the tower is in two properties of the category of exceptional covers of (Z,Fq ) of the normal variety Z:

One value of the tower construction is that it allows using subtowers consisting of the minimal subcategory closed under fiber products generated by particular exceptional covers. It is natural to associate subtowers to well-understood results, and equally natural to phrase mysteries in terms of other subtowers. For example, specific subtowers generated by genus 0 exceptional covers to organize results and unsolved problems from [FGS], [LMT] and [GMS]. The projective limit group is a surrogate for a universal covering space that works over finite fields.

A first characterization: You recognize exceptional covers from the first term of their extension of constants series, a purely Galois theoretic property that shows up as a Weil relation (their coefficients are the same at infinitely many terms) between the Poincare series associated to the two spaces in the cover. Such a Weil relation arises more generally if there is a chain of exceptional correspondences between two normal varieties over Fq. This gives an example of the type of Chow motive questions that exceptional covers and Davenport pairs (below) raise. Subexample: If for infinitely many t the t-th coefficient of the Poincare series of a curve over Fq is qt+1, is there a chain of exceptional correspondences from the projective line to the curve?

P(ossibly)r(educible)-exceptional covers generalize exceptional covers: Significantly different covers of a curve Z can have the same ranges on Fqt points for ∞-ly many t. We call these Davenport pairs and they produce pr-exceptional covers. An exceptional correspondence (over Fq) between two covers of Z  forces them to be an example of an isovalent Davenport pair (ranges have the same multiplicities). iDPs in general produce universal relations between Poincare series. We use two problems to show how relations arise in practice.

§4 considers the crytological uses of a given exceptional tower. Restrict to covers (degree at least 2) of projective r space Pr by itself. Then, you have the natural notion of the period of the map mt for those values of t in the exceptionality set of the map. RSA uses Euler's Theorem, to compute mt as n to the qt-1 power when r=1, and the map is the n-th power. A version of this also works for twists of Chebychev polynomials.

Serre's OIT: That raises the question of considering the same question for the two major examples of exceptional covers where r=1 from Serre's O(pen) I(mage) T(heorem). There are those of prime degree from the CM part of the theorem, and those of prime degree squared from the GL2 part of the OIT. The theory of complex multiplication gives a version of Euler's Theorem to handle the computation of periods in the first case.

The GL2 case is more exciting because you can start with one elliptic curve E over Q (given by a non-complex multiplication j invariant) and consider any particular prime n and the isogeny from multiplication by n on E. Excluding a finite set of nexceptions to the OIT – you can descend through Weierstrass normal form to a rational map fn: P1→ P1 with the following properties.

To get a sense of the explicit problems about cryptology periods and the description of these primes p, the paper consider Og's elliptic curve 3+ in the style mirroring Serre's use of it in his famous Chebotarev paper.